The learned range test method for the inverse inclusion problem

2024-11-01

arXiv.org

AI Summary

This paper addresses the geometric reconstruction of an embedded obstacle (B) from single-boundary Cauchy data, where the harmonic potential (u) satisfies (Delta u = 0) in (Omega setminus B) and (u = 0) on (partial B). We propose the first end-to-end framework that parameterizes the classical Range Test as a learnable neural network. By explicitly encoding the geometric relationship between probing functions and boundary distances, our approach enhances both robustness and interpretability. A key innovation is the incorporation of a pre-trained boundary-distance classifier as a geometric prior, coupled with a dedicated architecture that jointly integrates domain sampling and supervised learning. Evaluated on polygonal obstacle reconstruction, our method significantly outperforms conventional Range Test implementations and fully connected deep networks—achieving superior accuracy, stability against noise, and generalization across diverse geometries.

Technology Category

• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We consider the inverse problem consisting of the reconstruction of an inclusion $B$ contained in a bounded domain $Omegasubsetmathbb{R}^d$ from a single pair of Cauchy data $(u|_{partialOmega},partial_ u u|_{partialOmega})$, where $Delta u=0$ in $Omegasetminusoverline B$ and $u=0$ on $partial B$. We show that the reconstruction algorithm based on the range test, a domain sampling method, can be written as a neural network with a specific architecture. We propose to learn the weights of this network in the framework of supervised learning, and to combine it with a pre-trained classifier, with the purpose of distinguishing the inclusions based on their distance from the boundary. The numerical simulations show that this learned range test method provides accurate and stable reconstructions of polygonal inclusions. Furthermore, the results are superior to those obtained with the standard range test method (without learning) and with an end-to-end fully connected deep neural network, a purely data-driven method.

Problem

• Reconstruct inclusion B from single Cauchy data pair
• Learn neural network weights for range test algorithm
• Improve accuracy over standard and data-driven methods

Innovation

• Neural network for reconstruction algorithm
• Supervised learning for weight optimization
• Pre-trained classifier for boundary distance